Log₇(9 - X²): Domain Explained Simply

Hey guys! Ever wondered what makes a logarithmic function tick? Or, more specifically, what values you can actually plug into it without breaking the mathematical universe? Today, we're going to unravel the mystery behind the domain of the function log₇(9 - x²). We'll not only figure out the values of x that make this function work but also pinpoint those sneaky values that make it go undefined. So, buckle up, and let's dive into the fascinating world of logarithms!

What's the Domain, Anyway?

Before we jump into the specifics, let's quickly recap what the domain of a function actually means. Simply put, the domain is the set of all possible input values (in this case, 'x' values) that you can feed into a function and get a real number as an output. Think of it like this: the function is a machine, and the domain is the list of ingredients you can safely put in without causing a malfunction. For logarithmic functions, there are certain restrictions on what we can put in, which we'll explore in detail.

Logarithmic Functions: The Basics

Logarithmic functions are the inverse of exponential functions. The general form of a logarithmic function is logₐ(b), where 'a' is the base and 'b' is the argument. The logarithm answers the question: "To what power must we raise 'a' to get 'b'?" For instance, log₂(8) = 3 because 2³ = 8. However, there's a catch! Logarithmic functions have a very important rule: the argument (b) must be strictly greater than zero. This is because you can't raise any number to a power and get zero or a negative number.

Cracking the Code: Finding the Domain of log₇(9 - x²)

Now, let's get our hands dirty with the function log₇(9 - x²). Our mission is to find the values of 'x' that make this function valid. Remember that key rule we just talked about? The argument of the logarithm (in this case, 9 - x²) must be greater than zero. This gives us our starting point:

9 - x² > 0

This is an inequality, and to solve it, we need to do a little bit of algebraic maneuvering. Let's break it down step by step.

Step 1: Rearrange the Inequality

First, let's rearrange the inequality to make it easier to work with. We can add x² to both sides:

9 > x²

Alternatively, we can rewrite this as:

x² < 9

Step 2: Taking the Square Root

Now, we need to get 'x' by itself. To do this, we'll take the square root of both sides. But here's a crucial point: when you take the square root of both sides of an inequality, you need to consider both the positive and negative roots.

√(x²) < √9

This gives us:

|x| < 3

Step 3: Interpreting the Absolute Value

The absolute value inequality |x| < 3 means that the distance of 'x' from zero is less than 3. In other words, 'x' must be between -3 and 3. We can write this as:

-3 < x < 3

And there you have it! The domain of the function log₇(9 - x²) is all the values of 'x' that satisfy -3 < x < 3.

Values That Make the Function Indefinite

So, we've found the values of 'x' that work. But what about the values that don't? These are the values that make the argument of the logarithm zero or negative. Let's figure them out.

When 9 - x² is Zero

First, let's find the values of 'x' that make 9 - x² equal to zero:

9 - x² = 0

Add x² to both sides:

9 = x²

Take the square root of both sides:

x = ±3

So, when x = 3 or x = -3, the argument of the logarithm becomes zero, and the function is undefined.

When 9 - x² is Negative

Next, let's consider when 9 - x² is negative:

9 - x² < 0

Add x² to both sides:

9 < x²

Or, rewrite it as:

x² > 9

Taking the square root of both sides, we get:

|x| > 3

This means that 'x' is either less than -3 or greater than 3. In interval notation, this is x < -3 or x > 3. These values also make the function undefined because they result in a negative argument for the logarithm.

Putting It All Together

So, to recap:

• The domain of log₇(9 - x²) is -3 < x < 3. This is the interval where the function is happy and produces real number outputs.
• The function is undefined when x = 3, x = -3, x < -3, or x > 3. These are the values that cause our logarithmic machine to break down.

Analyzing the Alternatives

Now, let's look at the alternatives provided and see which one matches our findings:

A) x < -3 ou x > 3

B) -3 < x < 3

C) x = 3

D) x = -3

Based on our analysis, the correct answer is B) -3 < x < 3. This is the interval that represents the domain of the function log₇(9 - x²).

Why Understanding Domains Matters

You might be wondering,